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A covariant nonlocal Lagrangian for the description of the scalar kaonic sector

Mesons are extended objects, hence their interaction can be described by utilizing form factors. At the Lagrangian level, one can use nonlocal interaction terms. Here we describe two possible nonlocal Lagrangians leading to a 3D form factor: the first one is simple but does not fulfill covariance (if one insists on a 3D cutoff), the second extension is more involved but guarantees covariance. Such form factors are useful when calculating mesonic loops. As an important example, we discuss the scalar kaonic sector, $I(J^{P})=\frac{1}{2}(0^{+})$. The Lagrangian contains a single scalar kaon (the well-establish state $K_{0}^{\ast}(1430)$), but through loops $K_{0}^{\ast}(800)$ emerges as a dynamically generated companion pole (which disappears in the large-$N_{c}$ limit).